From [1]} and because \(\Vert Y\Vert =\sup _{\Vert u\Vert =1}\langle Y,u\rangle \) , there exists an absolute constant \(C_1\)  such that, for all \(t\ge 0\) ,
\(\mathbb {P}\left(\left\Vert \sum _{i=1}^n Y_i\right\Vert \ge \frac{3}{2}\mathbb {E}\left[\left\Vert \sum _{i=1}^n Y_i\right\Vert \right]+t \right)\le \exp \left( -\frac{t^2}{4n\sigma ^2}\right)+3\exp \left(-\frac{t}{C_1\Vert  \max _{1\le i\le n} \Vert Y_i\Vert  \Vert _{\psi _1}} \right). \) 
